a^2 - b^2 = (a+b)(a-b) = 10000000^2 = 10^14 = (2^14)*(5^14).
We want (a+b) to be as big as possible, which means (a-b) should be as small as possible. As we can see by the factorization of it, (a-b) must be at least 2.
So we have a-b = 2 and a+b = (2^13)*(5^14).
a = b + 2 --> 2b + 2 = (2^13)*(5^14) --> b + 1 = (2^12)*(5^14) --> b = (2^12)*(5^14) - 1 = 24999999999999. a = 25000000000001.